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Mattia Magnabosco

Publications and source records attributed to Mattia Magnabosco.

At least 19 recordsLinked to original sources
arXiv2026

New Topological Restrictions For Spaces With Nonnegative Ricci Curvature

We obtain new topological restrictions for complete Riemannian manifolds with nonnegative Ricci curvature and RCD(0,n) spaces. Our main results are a Betti number rigidity theorem which answers a question open since work of M.-T. Anderson in 1990, and a vanishing theorem for the simplicial volume generalizing a theorem of M. Gromov from 1982. Combining such results we obtain a new proof of the classification of noncompact 3-manifolds with nonnegative Ricci curvature, originally due to G. Liu in 2011, which extends to the synthetic setting.

math.DG↗
arXiv2025

Curvature exponent of sub-Finsler Heisenberg groups

The curvature exponent $N_{\mathrm{curv}}$ of a metric measure space is the smallest number $N$ for which the measure contraction property $\mathsf{MCP}(0,N)$ holds. In this paper, we study the curvature exponent of sub-Finsler Heisenberg groups equipped with the Lebesgue measure. We prove that $N_{\mathrm{curv}} \geq 5$, and the equality holds if and only if the corresponding sub-Finsler Heisenberg group is actually sub-Riemannian. Furthermore, we show that for every $N\geq 5$, there is a sub-Finsler structure on the Heisenberg group such that $N_{\mathrm{curv}}=N$.

math.MG↗
arXiv2025

Gradient flows of $(K,N)$-convex functions with negative $N$

We discuss $(K,N)$-convexity and gradient flows for $(K,N)$-convex functionals on metric spaces, in the case of real $K$ and negative $N$. In this generality, it is necessary to consider functionals unbounded from below and/or above, possibly attaining as values both the positive and the negative infinity. We prove several properties of gradient flows of $(K,N)$-convex functionals characterized by Evolution Variational Inequalities, including contractivity, regularity, and uniqueness.

math.FA↗
arXiv2025

On the rectifiability of $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ spaces with unique tangents

We prove rectifiability results for $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ metric measure spaces $(\mathsf{X},\mathsf{d},\mathfrak{m})$ with pointwise Ahlfors regular reference measure $\mathfrak{m}$ and with $\mathfrak{m}$-almost everywhere unique metric tangents. In particular, we show rectifiability if (i) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{CD}(K,N)$ for an arbitrary $N$ and has Hausdorff dimension $n<5$, or (ii) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{MCP}(K,N)$ and non-collapsed, namely it has Hausdorff dimension $N$. Our strategy is based on the failure of the $\mathsf{CD}$ condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed $\mathsf{MCP}$ on sub-Finsler Carnot groups, and on the recent breakthrough by Bate [Invent. Math., 230(3):995-1070, 2022].

math.MG↗
arXiv2025

On perimeter minimizing sets in manifolds with quadratic volume growth

This paper studies whether the presence of a perimeter minimizing set in a Riemannian manifold $(M,g)$ forces an isometric splitting. We show that this is the case when $M$ has non-negative sectional curvature and quadratic volume growth at infinity. Moreover, we obtain that the boundary of the perimeter minimizing set is identified with a slice in the product structure of $M$.

math.DG↗
arXiv2025

A review of the tangent space in sub-Finsler geometry and applications to the failure of the $\mathsf{CD}$ condition

We review the construction of the tangent space to a sub-Finsler manifold in the measured Gromov-Hausdorff sense. Under suitable assumptions on the measure, the metric measure tangent is described by the nilpotent approximation, equipped with a scalar multiple of the Lebesgue measure. We apply this result in the study of the Lott-Sturm-Villani curvature-dimension condition in sub-Finsler geometry. In particular, we show the failure of the $\mathsf{CD}$ condition in 3D-contact sub-Finsler manifolds, equipped with a bounded measure.

math.DG↗
arXiv2024

Metric conditions that guarantee existence and uniqueness of Optimal Transport maps

We investigate metric conditions that allow to prove existence and uniqueness of a map solving the Monge problem between two marginals in a metric (measure) space, proving two main results. Firstly, we introduce a nonsmooth version of the Riemannian twist condition that we call local metric twist condition, showing, under this assumption on the cost function, existence and uniqueness of optimal transport maps. Secondly, we prove the same result for cost equal to $d^2$ in a metric space $(X, d)$ satisfying a quantitative non-branching assumption, that we call locally-uniformly non-branching.

math.MG↗
arXiv2024

On the existence of $L^p$-Optimal Transport maps for norms on $\mathbb{R}^N$

In this paper, we prove existence of $L^p$-optimal transport maps with $p \in (1,\infty)$ in a class of branching metric spaces defined on $\mathbb{R}^N$. In particular, we introduce the notion of cylinder-like convex function and we prove an existence result for the Monge problem with cost functions of the type $c(x, y) = f(g(y - x))$, where $f: [0, \infty) \rightarrow [0, \infty)$ is an increasing strictly convex function and $g: \mathbb{R}^N \rightarrow [0, \infty)$ is a cylinder-like convex function. When specialised to cylinder-like norm, our results shows existence of $L^p$-optimal transport maps for several "branching'" norms, including all norms in $\mathbb{R}^2$ and all crystalline norms.

math.MG↗
arXiv2024

Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups

In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short $\mathsf{MCP}$, proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither $C^1$ nor strongly convex, the associated Heisenberg group does not satisfy $\mathsf{MCP}(K,N)$ for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a $C^{1,1}$ and strongly convex norm, and with the Lebesgue measure, satisfies $\mathsf{MCP}(0,N)$ for some $N \in (1,\infty)$. Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of $C^1$ and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short $\mathsf{CD}(K,N)$. We show that the sub-Finsler Heisenberg group, equipped with a $C^1$ and strongly convex norm, and with a positive smooth measure, does not satisfy the $\mathsf{MCP}(K,N)$ condition for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. Combining this result with our findings regarding the measure contraction property, we conclude the failure of the $\mathsf{CD}$ condition in the Heisenberg group for every sub-Finsler structure.

math.MG↗
arXiv2023

Failure of the curvature-dimension condition in sub-Finsler manifolds

The Lott-Sturm-Villani curvature-dimension condition $\mathsf{CD}(K,N)$ provides a synthetic notion for a metric measure space to have curvature bounded from below by $K$ and dimension bounded from above by $N$. It has been recently proved that this condition does not hold in sub-Riemannian geometry for every choice of the parameters $K$ and $N$. In this paper, we extend this result to the context sub-Finsler geometry, showing that the $\mathsf{CD}(K,N)$ condition is not well-suited to characterize curvature in this setting. Firstly, we show that this condition fails in (strict) sub-Finsler manifolds equipped with a smooth strongly convex norm and with a positive smooth measure. Secondly, we focus on the sub-Finsler Heisenberg group, proving that curvature-dimension bounds can not hold also when the reference norm is less regular, in particular when it is of class $C^{1,1}$. The strategy for proving these results is a non-trivial adaptation of the work of Juillet [Rev. Mat. Iberoam., 37(1):177-188, 2021], and it requires the introduction of new tools and ideas of independent interest. Finally, we demonstrate the failure of the (weaker) measure contraction property $\mathsf{MCP}(K,N)$ in the sub-Finsler Heisenberg group, equipped with a singular strictly convex norm and with a positive smooth measure. This result contrasts with what happens in the \sr Heisenberg group, which instead satisfies $\mathsf{MCP}(0,5)$.

math.MG↗
arXiv2023

Examples of $CD(0,N)$ spaces with non-constant dimension

In this work, we generalize the results obtained in (J. Geom. Anal., 32(6):Paper No.173, 32, 2022), presenting some examples of $CD(0,N)$ spaces having different dimensions in different regions, deducing in particular that the topological splitting may fail in $CD(0,N)$ spaces. We also observe that any reasonable non-branching condition may fail in $CD(0,N)$ spaces and that the existence of an optimal transport map, between two absolutely continuous marginals, is not guaranteed by the $CD(0,N)$ condition, without requiring a non-branching assumption. Moreover, we show that the strict $CD(0,N)$ condition is strictly stronger than the classical $CD(0,N)$ one and it is not stable with respect to the measured Gromov-Hausdorff convergence.

math.MG↗
arXiv2023

Almost-Riemannian manifolds do not satisfy the $\mathsf{CD}$ condition

The Lott-Sturm-Villani curvature-dimension condition $\mathsf{CD}(K,N)$ provides a synthetic notion for a metric-measure space to have curvature bounded from below by $K$ and dimension bounded from above by $N$. It was proved by Juillet that a large class of \sr manifolds do not satisfy the $\mathsf{CD}(K,N)$ condition, for any $K\in\mathbb R$ and $N\in(1,\infty)$. However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the $\mathsf{CD}$ condition in this setting, providing a new strategy which allows us to contradict the $1$-dimensional version of the $\mathsf{CD}$ condition. In particular, we prove that $2$-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the $\mathsf{CD}(K,N)$ condition for any $K\in\mathbb R$ and $N\in(1,\infty)$.

math.DG↗
arXiv2022

The strong Brunn--Minkowski inequality and its equivalence with the CD condition

In the setting of essentially non-branching metric measure spaces, we prove the equivalence between the curvature dimension condition CD(K,N), in the sense of Lott--Sturm--Villani, and a newly introduced notion that we call strong Brunn--Minkowski inequality SBM(K,N). This condition is a reinforcement of the generalized Brunn--Minkowski inequality BM(K,N), which is known to hold in CD(K,N) spaces. Our result is a first step towards providing a full equivalence between the CD(K,N) condition and the validity of BM(K,N), which we have recently proved in the framework of weighted Riemannian manifolds.

math.MG↗
arXiv2022

The Brunn--Minkowski inequality implies the CD condition in weighted Riemannian manifolds

The curvature dimension condition CD(K,N), pioneered by Sturm and Lott--Villani, is a synthetic notion of having curvature bounded below and dimension bounded above, in the non-smooth setting. This condition implies a suitable generalization of the Brunn--Minkowski inequality, denoted by BM(K,N). In this paper, we address the converse implication in the setting of weighted Riemannian manifolds, proving that BM(K,N) is in fact equivalent to CD(K,N). Our result allows to characterize the curvature dimension condition without using neither the optimal transport nor the differential structure of the manifold.

math.DG↗
arXiv2022

A canonical infinitesimally Hilbertian structure on locally Minkowski spaces

The aim of this paper is to show the existence of a canonical distance $\mathsf d'$ defined on a locally Minkowski metric measure space $(\mathsf X,\mathsf d,\mathfrak m)$ such that: i) $\mathsf d'$ is equivalent to $\mathsf d$, ii) $(\mathsf X, \mathsf d', \mathfrak m)$ is infinitesimally Hilbertian. This new regularity assumption on $(\mathsf X, \mathsf d,\mathfrak m)$ essentially forces the structure to be locally similar to a Minkowski space and defines a class of metric measure structures which includes all the Finsler manifolds, and it is actually strictly larger. The required distance $\mathsf d'$ will be the intrinsic distance $\mathsf d_\mathsf{KS}$ associated to the so-called Korevaar-Schoen energy, which is proven to be a quadratic form. In particular, we show that the Cheeger energy associated to the metric measure space $(\mathsf X, \mathsf d_\mathsf{KS}, \mathfrak m)$ is in fact the Korevaar-Schoen energy.

math.MG↗
arXiv2021

Convergence of metric measure spaces satisfying the CD condition for negative values of the dimension parameter

We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we introduce an appropriate setting to introduce the CD(K, N)-condition for $N < 0$, allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we introduce the distance $d_{\mathsf{iKRW}}$, which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD(K, N)-condition with $N < 0$ is converging with respect to the distance $d_{\mathsf{iKRW}}$ to some metric measure space, then this limit structure is still a CD(K, N) space.

math.MG↗
arXiv2021

Example of an Highly Branching CD Space

Ketterer and Rajala showed an example of metric measure space, satisfying the measure contraction property $MCP(0,3)$, that has different topological dimensions at different regions of the space. In this article I propose a refinement of that example, which satisfies the $CD(0,\infty)$ condition, proving the non-constancy of topological dimension for CD spaces. This example also shows that the weak curvature dimension bound, in the sense of Lott-Sturm-Villani, is not sufficient to deduce any reasonable non-branching condition. Moreover, it allows to answer to some open question proposed by Schultz, about strict curvature dimension bounds and their stability with respect to the measured Gromov Hausdorff convergence.

math.MG↗